In this paper, powerful induced-cyclic totally decomposed cumulative Goppa (TDCG) codes are selected and used for coded-cooperative diversity, which can fit wireless communication applications. Two TDCG codes employed at the source and relay nodes establish the joint design of the destination channel code. In the cooperation phase, we consider an optimized relay channel code which improves the overall bit-error-rate (BER) performance. Two methods of optimization are explored. The most practical method for optimization in terms of reduced processing complexity is proposed. We use a joint decoding which exploits all the resources gained by the cooperation scheme at the destination. The BER performance is evaluated through Monte-Carlo simulations with BPSK modulation and coherent detection over the flat Rayleigh fading channel. The proposed TDCG coded-cooperative diversity scheme exhibit a better BER performance than the non-cooperation scheme and other existing coded cooperation methods of similar code rates under similar channel conditions.

This paper presents a multi-relay Coded-Cooperative system that integrates Hybrid Automatic Repeat Request and Coded Cooperative Cooperation. It utilizes powerful cyclic Totally Decomposed Cumulative Goppa codes (TDCG codes) known for exceptional performance as best-known linear codes. The destination code is jointly designed by combining TDCG codes from relay and source codes. Optimized relay channel codes improve overall system bit-error-rate performance, aided by a proposed optimization technique. Numerical simulations assess BER performance under various channel conditions with BPSK modulation and coherence detection over flat Rayleigh fading channels. Results show that the TDCG coded-cooperative scheme outperforms non-cooperative approaches and similar coded cooperation techniques with comparable code rates. The integration of HARQ and Coded Cooperative Cooperation enhances wireless communication system reliability and efficiency.

The classical Extended Euclidean Algorithm (EEA)-based decoder for binary Goppa codes is designed to correct up to \( \lfloor w/2 \rfloor \) errors, where \( w \) is the degree of the Goppa polynomial. When the number of errors exceeds this correction capability, the EEA decoding output may contain non-binary symbols, resulting in an invalid estimate of the transmitted binary codeword. This paper proposes an EEA-based decoding framework enhanced by a localized maximum-likelihood (LML) refinement step, referred to as the EEA-LML decoder. In the proposed method, the received vector \( r\in\mathbb{F}_2^n \) is first decoded using the EEA decoder to obtain a tentative estimate \( \hat{\mathbf{c}} \). If \( \hat{\mathbf{c}} \) contains non-binary components, these positions are treated as ambiguous and resolved through a localized likelihood-based search subject to the syndrome constraint, yielding a final binary estimate \( \bar{\mathbf{c}}\in\mathbb{F}_2^n \). Monte Carlo simulations are conducted over additive white Gaussian noise (AWGN) and flat Rayleigh fading channels using BPSK modulation. The bit error rate (BER) performance of the proposed EEA-LML decoder is compared with the locator-only EEA decoder, maximum-likelihood decoding (MLD), and Patterson decoding for selected binary Goppa codes. The simulation results show that the proposed EEA-LML decoder improves the BER performance relative to the locator-only EEA decoder, while maintaining a more practical decoding structure than exhaustive MLD. The comparison with Patterson decoding further illustrates the potential benefit of the localized refinement step in improving the reliability of EEA-based decoding beyond the conventional correction radius.